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Defining Exponential and Trigonometric Functions Using Differential Equations

Harvey Diamond
Mathematics Magazine
Vol. 87, No. 1 (February 2014), pp. 37-42
DOI: 10.4169/math.mag.87.1.37
Stable URL: http://www.jstor.org/stable/10.4169/math.mag.87.1.37
Page Count: 6
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Defining Exponential and Trigonometric Functions Using Differential Equations
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Abstract

Summary This note addresses the question of how to rigorously define the functions exp(x), sin(x), and cos(x), and develop their properties directly from that definition. We take a differential equations approach, defining each function as the solution of an initial value problem. Assuming only the basic existence/uniqueness theorem for solutions of linear differential equations, we derive the standard properties and identities associated with these functions. Our target audience is undergraduates with a calculus background.

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